Maximum likelihood thresholds via graph rigidity
Maximum likelihood thresholds via graph rigidity
复制标题
通过图刚性确定最大似然阈值
DOI：
10.1214/23-aap2039
复制
发表时间：
2021
期刊：
影响因子：
--
通讯作者：
Louis Theran
中科院分区：
文献类型：
--
作者：
D. Bernstein;Sean Dewar;S. Gortler;A. Nixon;Meera Sitharam;Louis Theran
图G$的最大似然阈值（MLT）是在相应的高斯图模型中几乎肯定保证最大似然估计存在的最小样本数。我们给出了一个新的特征的MLT的刚性理论性质的$G$，并使用此特征给出新的组合下界的MLT的任何图。我们使用新的下界给稀疏Erd{\“o}s-R\'enyi随机图的最大似然阈值的平均密度方面的高概率保证。这些例子表明，新的下界是在一个polylog因子紧，其中，在同一个图的家庭，所有已知的下界是平凡的。基于我们的方法所做的计算实验，我们猜想Erd{\“o}s-R\'enyi随机图的MLT以高概率等于它的一般完备秩.利用低维刚性图的结构性结果，我们证明了图的最大极限为4的猜想，并描述了图的最大极限从3变为4的阈值概率.我们还给出了一个新的“提升”问题的框架，这是有趣的在自己的权利方面的图形的MLT的几何特征。提升的角度产生了一个新的连接弱MLT（其中最大似然估计只存在正概率）和经典的Hadwiger-Nelson问题。
The maximum likelihood threshold (MLT) of a graph $G$ is the minimum number of samples to almost surely guarantee existence of the maximum likelihood estimate in the corresponding Gaussian graphical model. We give a new characterization of the MLT in terms of rigidity-theoretic properties of $G$ and use this characterization to give new combinatorial lower bounds on the MLT of any graph. We use the new lower bounds to give high-probability guarantees on the maximum likelihood thresholds of sparse Erd{\"o}s-R\'enyi random graphs in terms of their average density. These examples show that the new lower bounds are within a polylog factor of tight, where, on the same graph families, all known lower bounds are trivial. Based on computational experiments made possible by our methods, we conjecture that the MLT of an Erd{\"o}s-R\'enyi random graph is equal to its generic completion rank with high probability. Using structural results on rigid graphs in low dimension, we can prove the conjecture for graphs with MLT at most $4$ and describe the threshold probability for the MLT to switch from $3$ to $4$. We also give a geometric characterization of the MLT of a graph in terms of a new"lifting"problem for frameworks that is interesting in its own right. The lifting perspective yields a new connection between the weak MLT (where the maximum likelihood estimate exists only with positive probability) and the classical Hadwiger-Nelson problem.
影响因子：
1.1
作者：
D. Bernstein;Grigoriy Blekherman;Rainer Sinn
通讯作者：
D. Bernstein;Grigoriy Blekherman;Rainer Sinn
DOI：
10.1214/18-aos1724
发表时间：
2019
期刊：
The Annals of Statistics
影响因子：
--
作者：
Drton, Mathias;Fox, Christopher;Käufl, Andreas;Pouliot, Guillaume
通讯作者：
Pouliot, Guillaume